How do I get into edit mode as shown in this video?
You have added the modifiers, but you have not applied them yet. That's why it looks different. The transparent cube on your screenshot is the unmodified mesh. The sphere is the result of the Subdivision Surface modifier. The Displacement modifier is disabled in Edit Mode (see icons of the modifiers).
In Blender 2.9x the apply button of modifiers has been removed. It's now hidden in a drop down menu. Apply the modifiers and it should look like in the video.